a survey on multivariate data...

A Survey on Multivariate Data Visualization

Winnie Wing-Yi Chan

Department of Computer Science and Engineering

Hong Kong University of Science and Technology

Clear Water Bay, Kowloon, Hong Kong

June 2006

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Table of Contents

Table of Contents 2

Abstract 4

1 Introduction 5

1.1 Motivations………………………………………………………………… 5

1.2 Challenges…………………………………………………………………. 5

2 Concepts and Terminology 6

2.1 Dimensionality……………………………………………………………... 6

2.2 Multidimensional and Multivariate………………………………………… 8

3 Visualization Techniques 8

3.1 Classifications……………………………………………………………… 8

3.2 Geometric Projection………………………………………………………. 8

3.2.1 Scatterplot Matrix………………………………………………… 9

3.2.2 Prosection Matrix………………………………………………… 10

3.2.3 HyperSlice………………………………………………………… 10

3.2.4 Hyperbox………………………………………………………… 11

3.2.5 Parallel Coordinates……………………………………………… 11

3.2.6 Radial Coordinate Visualization………………………………….. 12

3.2.7 Andrews Curve…………………………………………………… 12

3.2.8 Star Coordinates…………………………………………………… 12

3.2.9 Table lens…………………………………………………………. 13

3.3 Pixel-Oriented Techniques…………………………………………………. 13

3.3.1 Space Filling Curve……………………………………………... 14

3.3.2 Recursive Pattern………………………………………………… 15

3.3.3 Spiral and Axes Techniques……………………………………… 15

3.3.4 Circle Segment…………………………………………………… 16

3.3.5 Pixel Bar Chart…………………………………………………… 16

3.4 Hierarchical Display……………………………………………………….. 17

3.4.1 Hierarchical Axis………………………………………………… 17

3.4.2 Dimensional Stacking……………………………………………. 18

3.4.3 Worlds Within Worlds……………………………………………. 18

3.4.4 Treemap…………………………………………………………… 19

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3.5 Iconography………………………………………………………………… 19

3.5.1 Chernoff Faces…………………………………………………….. 19

3.5.2 Star Glyph………………………………………………………… 20

3.5.3 Stick Figure……………………………………………………….. 20

3.5.4 Shape Coding…………………………………………………….. 21

3.5.5 Color Icon………………………………………………………… 21

3.5.6 Texture……………………………………………………………. 22

4 Discussion and Conclusion 25

Bibliography 26

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Abstract

Multivariate data visualization, as a specific type of information visualization, is an active

research field with numerous applications in diverse areas ranging from science communities

and engineering design to industry and financial markets, in which the correlations between

many attributes are of vital interest.

In this survey, we will first review the motivations and challenges of multivariate data

visualization. In section 2, a brief terminology is introduced. Some established techniques for

multivariate data visualization are described in section 3. These techniques are classified into

several categories to provide a basic taxonomy of the field. At the end of this survey, we will

discuss some future research directions.

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1. Introduction

1.1 Motivations

While information is growing in an exponential way, our world is flooded with data which,

we believe, should contain some kind of valuable information that can possibly expand the

human knowledge. However, extracting the meaningful information is a difficult task when

large quantities of data are presented in plain text or traditional tabular form. Effective

graphical representations of the data thus enjoy popularity by harnessing the human’s visual

perception capabilities.

Information visualization is the use of computer-based interactive visual representations

of abstract and non-physically based data to amplify human cognition. It aims at helping users

to effectively detect and explore the expected, as well as discovering the unexpected to gain

insight into the data. For multivariate data visualization, the dataset to be visually analyzed is

of high dimensionality and these attributes are correlated in some way.

Multivariate data are encountered in all aspects by researchers, scientists, engineers,

manufacturers, financial managers and various kinds of analysts. Multivariate data

visualization is hence strongly motivated by the many situations when they are trying to

obtain an integrated understanding of the data distributions and investigate the

inter-relationships between different data attributes. Such an effective visual display tool is

demanded to facilitate users to identify, locate, distinguish, categorize, cluster, rank, compare,

associate or correlate the underlying data [3].

1.2 Challenges

Multivariate data visualization faces the same challenges as information visualization does:

Finding good visual representations of a problem can be hard and undeterministic. In addition,

multivariate data poses problems in encoding its attributes in a single visual display.

Mapping. Finding a suitable mapping of high-dimensional multivariate data into a

2D visual form is never a simple task. It usually depends on the nature of datasets to

be visualized and is more related to human perception. Also, association of data

attributes to graphical entities requires extreme caution to avoid overwhelming the

observer’s viewing ability. Conjunction of several elements in the representations

may induce cognition overload to the users [6] and graphical attributes should

therefore be carefully selected such that they are easy to untangle. It is important

that different attributes can be viewed holistically for integrated analysis and, at the

same time, each dimension can be judged by users separately and independently.

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Dimensionality. Multivariate data is often of huge size and high dimensionality that

will most likely result a dense structure. It is hence difficult to present such data in a

single visual display, making it challenging to enable users to explore the data space

intuitively and interactively, as well as discriminating individual dimensions. Dual

view and distortion skills like fisheyes may be helpful to solve this problem.

Furthermore, the ordering of dimensions has a major impact on the expressiveness

of visualization [7]. Different arrangement allows different conclusions to be drawn,

but no ordering principle is established so far.

Design Tradeoffs. Visualization can provide a qualitative overview of large and

complex datasets so that users can look for structure, features, patterns, trends and

relationships more effectively [4]. Due to the high dimensionality of multivariate

data, we inevitably sacrifice the ability to show the details of each attributes [1] as

we have fewer graphic attributes for encoding. This situation may not be flavored

when quantitative analysis is required. For multivariate data visualization, there is

always a tradeoff between amount of information, simplicity and accuracy.

Assessment of Effectiveness. The ultimate goal of multivariate data visualization is

to gain insight into the data and show the possible correlation between different

attributes. In most cases certain correlations are not yet discovered prior to looking

at the visual display, and they are exactly what we want to acquire after visualization.

It is a paradox [5] that prohibits the assessment of effectiveness of an information

visualization technique: We do not know what valuable knowledge is present in the

data, so we hope to gain insight by visualizing it. Nevertheless, if we known nothing

about the pattern or relationship to be shown in the data representation, we can

never assess the effectiveness of a particular visualization technique.

2. Concepts and Terminology

2.1 Dimensionality

Dimensionality of a problem in information visualization refers to the number of attributes, or

more generally as variables, that presents in the data to be visualized [2]. For one-dimensional

data, which is also known as univariate data, consists of only one attributes, such as a

collection of houses characterized by the cost. They can be visualized effectively by

traditional tools like table and histogram. Interpretation of two-dimensional or bivariate data

usually utilizes the x-y coordinates of a 2D space. A conventional approach is to plot one

variable against the other called scatterplot, see Figure 2.1.

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Figure 2.1: A scatterplot illustrating wine consumption against deaths from heart disease. [8]

Technically, multivariate data, also termed hypervariate data, is defined for a high

dimensionality of three or above. However, as three-dimensional space are what we are living

in, three-dimensional or trivariate data is often entertained separately. Modeling the data in a

3D space is the most straightforward way, but problems arise with displaying it in a

two-dimensional representation [2]. It is hard to compare two points along the same axis, see

Figure 2.2(a). A feasible solution, as shown in Figure 2.2(b), is to project the points onto pairs

of axes in a two-dimensional scatterplot. 3D surfaces such as Figure 2.3(a) also encountered

the same difficulty [2], where the minimum value can only be obtained after altering the view

as in Figure 2.3(b). Obviously, orientation becomes crucial when dimensionality increases

and proper interaction should be able to tackle this problem.

(a)

(b)

Figure 2.2: (a) A 3D scatterplot, (b) Projection of the points in (a) onto two of the axes [9].

(a)

(b)

Figure 2.3: (a) A 3D surface, (b) A view of (a) by changing the orientation [10].

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The conceptual boundary between low and high dimensionality is not always precisely

stated [11]. High-dimensional data is used in a loose manner; it can be arbitrarily defined, but

it usually depicts a dimensionality of more than four. It is important to observe that geometric

projections in more than four-dimensional are ineffective to convey information to human,

which is due to the significant differences to perceive between low and high dimensionality.

2.2 Multidimensional and Multivariate

The terms multidimensional and multivariate are often used vaguely. Strictly speaking,

multidimensional refers to the dimensionality of the independent dimensions while

multivariate refers to that of the dependent variables [12]. The more appropriate term for

multivariate data visualization should be multidimensional multivariate data visualization

[13]. Nevertheless, a set of multivariate data is in high dimensionality and can possibly be

regarded as multidimensional because the key relationships between the attributes are

generally unknown in advance. The multidimensional property is therefore implied in

common usage.

For convenience, the term attributes denote both independent dimensions and dependent

variables. It also worth noting that multivariate data visualization is rather generic and does

not categorize itself clearly between information visualization and scientific visualization.

3. Visualization Techniques

3.1 Classifications

Keim and Kriegel [14] [15] divided visual data exploration techniques for multidimensional

multivariate data into six classes, namely geometric, icon-based, pixel-oriented, hierarchical,

graph-based and hybrid techniques. We will adopt this taxonomy and tailor it to multivariate

data visualization techniques, which are classified into four broad categories according to the

overall approaches taken to generate resulting visualizations [11]: Geometric projection,

pixel-oriented techniques, hierarchical display and iconography. They are elaborated in the

following sections. Some representative techniques in each group are described in detail.

3.2 Geometric Projection

Geometric projection techniques aim at finding informative projections and transformations

of multidimensional datasets [14]. It may map the attributes to a typical Cartesian plane like

scatterplot, or more innovatively to an arbitrary space such as parallel coordinates.

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Methods fall in this category are good for detecting outliers and correlation amongst

different dimensions, and handling huge datasets when appropriate interaction techniques are

introduced [15]. Intrinsically all data attributes are treated equally, but we must be aware that

all dimensions may not be perceived equally [2]. As the order in which axes are displayed

affects our perception [14], rearrangement is important if the display should not be biased.

Another potential problem is visual cluttering and record overlapping [14] which overwhelms

the user’s perception capabilities due to the high dimensionality or the large size of the data.

Some typical techniques using geometric projection are discussed next.

3.2.1 Scatterplot Matrix

Scatterplot is used for bivariate discrete data in which two attributes are projected along the

x-y axes of the Cartesian coordinates. Scatterplot matrix is an extension for multidimensional

data where a collection of scatterplots is organized in a matrix simultaneously to provide

correlation information among the attributes, see Figure 3.1. We can easily observe patterns in

the relationships between pairs of attributes from the matrix, but there may be important

patterns in higher dimensions which are barely recognized in it [17]. Another limitation is that

it becomes chaotic when the number of points, that is the number of data items, is too large.

Figure 3.1: A scatterplot matrix for 5-dimensional data of 400 automobiles [17].

Fortunately the technique of brushing [18] can be applied to address the above problem.

Brushing aims interpretation by highlighting a particular n-dimensional subspace in the

visualization [13], that is, the respective points of interested are colored or highlighted in each

scatterplot in the matrix. In Figure 3.1, automobiles are color-coded by the number of

cylinders. Manufacturers can analyze the performance of the cars based on the number of

cylinders for improvements, while customers can decide how many cylinders they need in

order to suit their needs.

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3.2.2 Prosection Matrix

Prosection was first introduced by Furnas and Buja [19]; Tweedie and Spence [20] later

extended it to prosection matrix which supports a higher dimensionality. A typical prosection

is shown in Figure 3.2(a). In the simplest sense, prosection is the orthogonal projections

where the data items lie in the selected multidimensional range are colored differently [15].

The yellow rectangles in Figure 3.2(b) indicate the tolerances on parameter values, which is

particularly useful for manufacturers to select appropriate parameter ranges. Yet it gives less

information about the correlations between more than two attributes.

Figure 3.2: (a) A prosection, (b) A prosection matrix [21].

3.2.3 HyberSlice

Like the scatterplot and prosection matrix, HyperSlice [22] has a matrix graphics representing

a scalar function of the variables [23], see Figure 3.3. This method targets at continuous scalar

functions rather than discrete data. The most significant improvement over scatterplot is the

interactive data navigation around a user defined focal point [23]. An enhanced HyperSlice

was also proposed [24] which incorporate the concept of display resolution supported by

space projection, together with the concept of data resolution provided by wavelets to form a

powerful multiresolution visualization system.

(a)

(b)

Figure 3.3: (a) Effect of dragging a slice [22], (b) HyperSlice for 4D function [23].

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3.2.4 Hyberbox

Hyperbox [25] works similarly with the above techniques, except that the plots are now

constructed as n-dimensional box instead of a matrix, as shown in Figure 3.5. The box is

depicted in two dimensional because it is impossible to model the box exactly in an

n-dimensional space. Hyberbox is a more powerful tool as it is possible to map variables to

both size and shape of the face. It also allows emphasizing or de-emphasizing some variables

[23]. However, the length and orientation are arbitrary which may convey the wrong

information as it violates the “banking to 45 degrees” principle [26].

Figure 3.5: (a) A hyberbox [23]. Figure 3.6: Parallel coordinates [17].

3.2.5 Parallel Coordinates

Parallel coordinates [27] [28] [29] is a well-know technique where attributes are represented

by parallel vertical axes linearly scaled within their data range. Each data item is represented

by a polygonal line that intersects each axis at respective attribute data value, see Figure 3.6.

Parallel coordinates can be used to study the correlations among attributes by spotting

the locations of the intersection points [23]. Also, they are effective for revealing the data

distributions and functional dependencies. Nevertheless, one major limitation is the limited

space available for each parallel axis. Visual clutter can severely hamper the user’s ability to

interpret and interact with the visualizations [11]. Similar problem arises when the

dimensionality of the data is too high that the axes are packed very closely. Same as the

previous techniques, brushing may be applied to aid interpretation.

Circular Parallel Coordinates [30] is one of the variations adopting a radial arrangement

of the axes, as illustrated in Figure 3.7. Hierarchical Parallel Coordinates [31] is an extension

that targets at large datasets. It displays the aggregation information derived from a

hierarchical clustering of the data [11]. These clusters are displayed at different levels of

abstraction with proximity-based coloring and structure-based brushing [32], see Figure 3.8.

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Figure 3.7: Circular Parallel coordinates [30].

Figure 3.8: Hierarchical Parallel Coordinates with different level of abstractions [31].

3.2.6 Andrews Curve

Andrews Curve [33], as shown in Figure 3.9, plots each data item as a curved line, which is

similar to a Fourier transform of a data point [30]. Close points result similar curves and

curves for distant points are distinct, which is useful for detecting clusters and outliers [34]. It

can cope with many dimensions but is computationally expensive to display large datasets.

3.2.7 Radical Coordinates Visualization

Radical Coordinates Visualization [30] is similar to parallel coordinates in spirit, in which n

lines emanate radically from the center of the circle and terminate at the perimeter, as shown

in Figure 3.10. Each line is associated with one attribute; spring constants attached to the data

attribute values define the positions of the data points along the lines. Points with

approximately equal or similar dimensional values lie closer to the center.

3.2.8 Star Coordinates

Star coordinates [35] is an extension of typical scatterplots to higher dimensions. Data items

are presented as points and attributes are represented by the axes arranged on a circle. Initially,

the angles between the axes are equal and all axes have the same length.

Users can apply scaling transformations to change the length of an axis, which increases

or decreases the contribution of an attribute. It also provide rotation transformations that

change the direction of an axis, so the angles are no more equal and thus making an attribute

more or less correlated with other attributes. An example of star coordinates after

transformation is shown in Figure 3.11.It has been found to be useful in gaining insight into

hierarchically clustered datasets and for multi-factor analysis for decision-making.

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Figure 3.9: Andrews Curves [30].

Figure 3.10: Radical Coordinates Visualization [30].

Figure 3.11: Star Coordinates with transformations [35].

3.2.9 Table Lens

In table lens [36], each row represents a data item and the columns refer to the attributes.

Each column is viewed as a histogram or as a plot, see Figure 3.12. Table lens was motivated

by the regularity nature of traditional tables, where information along rows or columns is

interrelated and can be interpreted as a coherent. It therefore takes advantage in using a

concept which we are familiar with. It allows users to spot relationships, analyze trends in

data, make assumptive correlations, easily view and manipulate the entire datasets.

Figure 3.12: An example of table lens from Inxight [37].

3.3 Pixel-Oriented Techniques

The second category for multivariate data visualization is pixel-oriented techniques. The idea

is to represent an attribute value by a pixel based on some color scale. For an n-dimensional

dataset, n colored pixels will be needed to represent one data item, with each attribute values

being placed in separate sub-windows, as illustrated in Figure 3.13.

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We can further divide these techniques into two subgroups, query-independent and

query-dependent. Query-independent techniques are favored by data with a natural ordering

according to one attribute, while query-dependent visualizations are more appropriate if the

feedback to some query is of interest [14]. For the latter, the distances of attribute values to

the query, instead of the absolute values, are mapped to colors. Correlations, functional

dependencies and other relationships between attributes may be detected by relating

corresponding regions in the multiple windows [14]. Moreover, as each data item is uniquely

mapped to a pixel, record overlap and visual cluttering are not likely [11].

Figure 3.13: Pixel-based visualization of 6-dimensional data [15].

3.3.1 Space Filling Curve

Space Filling Curves are query-independent that provides a better clustering of closely related

data items [14]. Some well-known examples are curves by Peano and Hilbert [38] [39] and

Morton [40]. For multivariate data, curves of particular attributes are display in separate

windows, as shown in Figure 3.14(a) and (b).

(a)

(b)

(c)

Figure 3.14: (a) Peano-Hibert, (b) Morton or Z-Curve, (c) Recursive Pattern [15].

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3.3.2 Recursive Pattern

Recursive Pattern [41] is another query-independent which is based on generic recursive

scheme to allow users to influence the arrangement of data items [14]. The arrangement of

lines and columns is performed iteratively and recursively; the elements to be arranged at

level i are the patterns resulting from level i-1. Similarly, recursive pattern of each attribute is

shown in a single window, as illustrated in Figure 3.14(c).

3.3.3 Spiral and Axes Techniques

Spiral and axes techniques are both query-dependent. Spiral technique [42] arranges the

pixels in spiral form according to the overall distance from the query, as depicted in Figure

3.15(a). The yellow center represents the data items satisfying the user specified query. Axes

technique [42] improves the spiral one by including feedback on displacement. Pixels are

arranged in partial spirals in each quadrant, that is, two attributes are assigned to the axes and

data items are arranged according to the displacement as shown in Figure 3.15(b).

For query-dependent techniques, an additional window, like the top left one in Figure

3.15(c) and (d), is provided for overall distance or displacement. By relating corresponding

regions in the windows, users can perceive multidimensional clusters or correlations [14].

(a)

(b)

i

j

(c)

(d)

Figure 3.15: (a) Spiral arrangement, (b) Partial spiral arrangement, (c) Spiral technique result

of an 8-dimensional dataset, (d) Axes technique result of the same dataset [15].

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3.3.4 Circle Segment

The design of circle segment [43] is to assign attributes on the segments of a circle. Data

items are arranged within a segment so that a single data item appears in the same position at

different segments [11]. The ordering and colors of the pixels are similarly determined by

their overall distance to the query. Examples of circle segments are shown in Figure 3.16.

(a)

(b)

Figure 3.16: (a) Circle segment arrangement for 8-diemensaionl data [15],

(b) An example of circle segments [7].

3.3.5 Pixel Bar Chart

Pixel bar chart [44], derived from regular bar chart, presents data values directly instead of

aggregating them into a few data values. Bars can be conventional histogram which plots one

attributes against its values as shown in Figure 3.17, or x-y diagram that plots one attribute

against another as illustrated in Figure 3.18. Each data item is represented by a single pixel

and is placed in the bars accordingly. Ordering within the bars is determined by two

additional attributes. Pixel color can be used to encode the values of one attributes.

For higher-dimensional data, multi-pixel bar charts are proposed, see Figure 3.18. Charts

are duplicated and different attribute is colored-coded for each chart. Thus the same data item

has the same relative position within each of the corresponding bars for detecting correlations.

(a)

(b)

Figure 3.17: (a) Equal-width pixel bar chart, (b) Equal-height pixel bar chart [44].

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Figure 3.18: Multi-pixel bar chart with color encoding different attributes [44].

3.4 Hierarchical Display

Hierarchical techniques subdivide the data space and present subspaces in a hierarchical

fashion [14]. Attributes are treated differently, with different mappings producing different

views of the underlying data. Therefore interpretation of resulting plots requires training [11].

The techniques lay in this category concern mainly hierarchical data, or data in which several

attributes are more important to users or of more interest.

3.4.1 Hierarchical Axis

In hierarchical axis [45] [46] [47], axes are laid out horizontally in a hierarchical fashion as

illustrated in Figure 3.19(a). This technique can plot many attributes in one screen [23]. One

simple example is the histograms within histograms plot. A matrix version, as shown in

Figure 3.19(b), is also introduced to enhance perception similar to scatterplot matrix.

(a)

(b)

Figure 3.17: (a) Hierarchical axes, (b) Histograms within histograms matrix [23].

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3.4.2 Dimensional Stacking

Dimensional stacking [48], also called general logic diagrams, is a variation of hierarchical

axis [23]. It partitions the data space into 2-dimensional subspaces which are stacked into

each other [15], as depicted in Figure 3.18. Those important attributes should be chosen for

the outer levels. This technique is especially adequate for discrete categorical or binned

ordinal values. A major advantage of dimensional stacking over hierarchical axis is that no

aggregation function is needed to plot the data, such as the previous case of histogram [23].

(a)

(b)

Figure 3.18: (a) Partition of dimensional stacking, (b) An example [15].

3.4.3 Worlds Within Worlds

Another well-know hierarchical technique is worlds within world, or n-vision [49]. The data

space is now subdivided into 3-dimenstional subspaces. It generates an interactive hierarchy

display, instead of the static objects in the previous one, by using powerful 3D rendering [23].

It allows the exploration of n-dimensional function spaces, but could also be adapted to

n-dimensional datasets [30]. Figure 3.19 shows an example that encodes 5-dimenisonal data.

Figure 3.19: N-Vision [50]. Figure 3.20: Treemap [52].

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3.4.4 Treemap

Treemap [51], as shown in Figure 3.20, uses a hierarchical partitioning of the screen into

regions, depending on the attribute values. The sizes of the nested rectangles represent the

attribute values, which provide extra information over simple mapping of dimensions only.

The color of the regions may encode an additional attribute. Treemap is suitable to obtain an

overview on large datasets with multiple ordinal attributes [15]. Moreover, it subdivides the

display in a space-filling manner that fully utilizes the available display space [53].

3.5 Iconography

Iconographic or icon-based techniques map each multidimensional data item to an icon, or

more specifically a glyph. The visual features vary depending on the data attribute values [11].

Several graphical parameters are usually contained in an icon, which makes it possible to

handle multidimensional data. Besides, observations of graphical features are pre-attentive

which is welcomed by human. However, unlike geometric techniques that treat all the

dimensions equally, some features in glyphs are more salient than others [11], adjacent

elements are easier to be related and accuracy of perceiving different graphical attributes

varies between humans tremendously. It thereby introduces biases in interpreting the result.

3.5.1 Chernoff Faces

Chernoff face visualization [54] is probably the most famous in iconography. Two attributes

are mapped to the 2D position of a face and remaining attributes are mapped to its properties

of the face, for instance, the shape of nose, mouth, eyes and that of the face itself, as

illustrated in Figure 3.21. One of the shortcomings is that different visual features are not

quite comparable to each other [11]. It is also suggested that Chernoff faces can only visualize

a limited amount of data items [14]. One common issue to all multidimensional icons,

including Chernoff faces, is that the semantic relation to the task has significant impact on the

perceptive effectiveness [2]. Yet it will then very much depend on the application domain.

(a)

(b)

Figure 3.21: (a) Chernoff faces in various 2D positions [54], (b) Different facial features [55].

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3.5.2 Star Glyph

There are many variants in the glyph family for displaying multidimensional data; star plot

[56] is one of the most widely used glyphs. The dimensions are represented as equal angular

axes radiating from the center of a circle [30], with an outer line connecting the data value

points on each axis, as depicted in Figure 3.22(b). Each data item is presented by one star

glyph. They are helpful for multivariate datasets of moderate size, but their primary weakness

is that the display becomes overwhelming when the number of data items increases. Star plots

can be further combined with other glyphs to encode extra information, an example

incorporating the traditional box-and-whisker plots is shown in Figure 3.22(c).

(a)

(b)

(c)

Figure 3.22: (a) Construct a star plot [57], (b) Group of star glyphs [30], (c) Box plot stars [58].

3.5.3 Stick Figure

Stick figure [59] is another classical icon-based technique that again maps two attributes to

the display axes and the remaining to the rotation angle, length, thickness or color of the

limbs, as depicted in Figure 3.23(a). When the data items are relatively dense with respect to

the display dimensions, the packed icons exhibit some texture patterns that vary according to

the data features, which are detected by pre-attentive perception [14]. However, the visual

discernment of an important pattern is highly dependent upon the selection of an appropriate

graphical attribute. This selection process is therefore deterministic and can a bottleneck [23].

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(a)

(b)

(c)

Figure 3.23: (a) Stick figure family [23], (b) 5D image data using stick figures,

(c) Part of (b) in original size [60].

3.5.4 Shape Coding

Shape coding [61] visualizes data using small arrays of pixels. Each data item is represented

by one such array, and the pixels are mapped to a color scale according to the attribute values,

see Figure 3.24. Pixels in the array are placed in the form of square or rectangle and the arrays

are arranged successively in a line-by-line fashion [14]. These arrays can contain an arbitrary

number of pixels, making it possible for multidimensional data visualization.

(a)

(b)

Figure 3.24: (a) Shape coding array [15], (b) An example [61].

3.5.5 Color Icon

Color icon [62] is a combination of the pixel-based spiral axes and icon-based shape coding

techniques. Pixels are replaced by arrays of color fields that represent the attribute values

similar to shape coding, as illustrated in Figure 3.25(c). Color, shape, size, orientation,

boundaries and area sub-dividers can all be used to map the multidimensional data [23]. Color

icon therefore merges color, shape and texture perception for iconographic integration [62].

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(a)

(b)

(c)

Figure 3.25: (a) 5D image data using color icons, (b) Part of (a) in original size [60],

(c) Color icon scheme [15].

3.5.6 Texture

When large multidimensional data is presented in icons, such as stick figures and color icons

mentioned above, it produces some textures which allow users to gain insight into the overall

relationships between attributes, in addition to individual data items encoded by the icons

respectively. With the recent advance in texture synthesis techniques [63] [64], it is now

feasible to apply textures directly in multivariate data visualization.

Figure 3.26: Hand-crafted example using textures to visualize 3D data [66].

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Richly detailed and varying texture patterns have vast potential in visualizing

multivariate data. Ware and Knight [65] conducted a pioneer vision research on using texture

for information display. They identified three dominant visual dimensions of textures, namely

orientation, size and contrast. Nevertheless the dimensionality of visual texture is very high,

other dimensions include but not limit to hue, luminance, scale, regularity, periodicity,

directionality, homogeneity, transparency, fuzziness and level of abstraction.

Among all, Interrante [66] proposed to harness natural textures of multivariate data

visualization. Figure 3.26 shows a hand-crafted example encoding three-dimensional data

with one dimension of color and two dimensions of texture. Tang et al. [67] applied natural

near-regular textures to visualize weather data with multi-layer controllable texture synthesis.

Additional attributes can be mapped to the foreground texture that is overlaid upon the

background texture. Figure 3.27 shows one of the resulting visualizations.

Figure 3.27: Texture synthesis result for large regions in China [67].

Healey et al. have been working on methods for visualizing large, complex and

multidimensional datasets [68] [69] [70]. Apart from the previous approach harnessing natural

and photographic textures, they proposed from a completely different direction of using

nonphotorealistic textures with perceptually-based brush strokes [71]. They used painted

brush strokes, as shown in Figure 3.28, to represent multidimensional data elements. Each

data attribute is mapped to a specific nonphotorealistic property such as color, orientation,

coverage, size, coarseness and weight. The attributes values can thus be identified from the

different visual appearances of the brush strokes.

24

The major advantage of using textures in multivariate data visualization is that they

contain various visual dimensions that human can distinguish effectively and pre-attentively.

Besides, the outcomes are generally more engaging and aesthetic that are more attractive and

favorable, independent of what type of textures are being used. But problems remain in

finding a suitable mapping from data attributes to texture features. Contrast illusions are also

induced when we are comparing the scale and orientation of textures [1], which may cause

misperception of data.

Figure 3.28: Nonphotorealistic visualization of weather conditions [68].

25

4. Discussion and Conclusion

In this survey, we reviewed some important techniques for multivariate data visualization. In

section 1, we presented the motivations and challenges of visualizing high-dimensional

multivariate data. A brief terminology on the topic was introduced in section 2, with an

emphasis on dimensionality and the concept of multidimensional multivariate data. In section

3, we categorized multivariate data visualization techniques into four classes, namely

geometric projection, pixel-oriented techniques, hierarchical display and iconography, based

on a scheme proposed by Keim and Kriegel for general information visualization techniques.

We aimed at providing a comprehensive overview on these techniques, which is arguable on

the basis of their major advantages and limitations.

Geometric projection techniques were long established before the field of information

visualization actually emerged. We are too familiar with the Cartesian space; it does not

require us much effort to understand the representations of such techniques. However, it

becomes problematic when the dimensionality of the data increases, as we can only map three

dimensions to a 3D space.

Pixel-oriented techniques encode each data item as pixels. The corresponding pixels

appear at the same position in each respective window. With suitable rearrangement, user may

observe the inter-relationships between attributes, trends and patterns in the underlying data.

Hierarchical displays are derived from the fundamental concept of hierarchical trees. They are

very effective in visualizing hierarchical data, which is also their limitation. The outcomes of

pixel-oriented techniques and hierarchical display are not as straightforward as those of

geometric projection do; training may be required in order to understanding the visualizations.

Iconography uses a multidimensional icon, or glyph, as the unit of visualization. A glyph

has numerous graphical properties that data attributes can map to. When the glyphs, which are

essentially data items, are densely packed together, it produces some texture patterns. Users

are thus able to study the overall features and relationships in the data. While color has been

used extensively to encode an addition dimension, it may be wise to replace its role by

textures that obviously provide more graphical attributes for higher dimensional data.

With the information explosion in the last decade, people are now able to access huge

amount of data easily from the internet; at the same time companies and institutions keep

growing their large databases. Some kind of information, which could be financially,

academically or even personally useful, is certainly under the veil of the impersonal data.

A picture is worth a thousand words. We believe multivariate data visualization has it

vantage in helping us to gain insights into the terabyte data, as well as recognize the hidden

correlations between attributes, which is beneficial to individuals, organizations, and possibly

the society.

26

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